TCHEBYCHEFF RATIONAL APPROXIMATION IN THE DISK, ON THE CIRCLE, AND ON A CLOSED INTERVAL

Abstract

Suppose that the function is analytic in the disk and continuous in its closure. Let denote the best uniform approximation of by rational functions of degree at most . In 1965 Dolzhenko established that if then belongs to the Hardy space . The following converse of this result is obtained here: if , then . In combination with results of Peller, Semmes, and the author, this estimate yields, in particular, a description of the set of functions with , where 1$ SRC=http://ej.iop.org/images/0025-5734/61/1/A06/tex_sm_3193_img11.gif/> and .Bibliography: 38 titles.